Transitions and Transformations

The trimester schedule that our school uses has many benefits, including 68 minute periods.  This seems to be the perfect amount of time for me as a math teacher to give the right amount of weight to each part of a lesson. One major drawback, however, is that sometimes I’m not ready to let them go.  Maybe it is an overall resistance to change, but more likely I feel that after 13 weeks, my classes are starting to make some major progress down the persistent problem solving path.  And then I have to let them go.  There are very few students that end up in my class more than once per trimester since the majority of what I teach is College Algebra and Probability and Statistics (both one trimester courses).

In my college algebra course, I’ve seen the progression and improvement in their problem solving abilities, but I wanted to see what they would say if I asked how they thought they had progressed.

So, as the last question on their final exam, I asked them what skills developed in this course they felt were most valuable.  (Oddly enough, none of them said factoring a quadratic equation.) But they did say some things that helped solidify my approach to this class.  Most frequently, students mentioned that they have improved their problem solving skills.  I’m confident that they are referencing problem solving skills in relation to rich problems, since that is a majority of what we did in this class.  Another common comment had to do with multiple approaches to solving problems.  Many students mentioned that they didn’t realize how many different ways there are to approach a problem until these varieties were laid out by other students. I, too, improved my ability to look for and appreciate the diversity in problem solving strategies.

My favorite comment overall was from a girl who works very hard but who hesitates to share her ideas with the class.  She states, “I can problem solve without a set procedure and now I feel like I can solve anything.”  Bingo.  I hope that I can help build that confidence as I work to improve some of the methods I used in this next trimester.

 

The fabulous life of Megan Schmidt

I read an article recently that asked when being “busy” became the new black.

When we ask students how they are, we always get the same answer: tired. Teachers, on the other hand can always be counted on to tell you how BUSY they are. Sometimes, you’ll even be lucky enough to get a teacher to tell you, “busy, but good.” As if to say, “I want to be polite, but get the heck out of my room so I can get something done!”

I once asked myself, if I had extra hours in the day, would I use them to be more productive with what I have on my plate? Or would I find new projects to fill up the time? If I’m being honest with myself, I’d have to say it would be the latter.

My day begins at about 5:25 am after about 45 alarm snoozes. I unglue the two beagles that have suctioned themselves to my arms and feet during the night. My husband and I leave to drop our daughter off at Montessori at 6:15. I want to take this moment to acknowledge how blessed I am to work at the same school as my husband. Getting to drop our daughter off together is a priceless bonus. I’m always drawn to the different building sets and usually sit down and play with her before we head off to our own school.

We try to arrive a little before 7am which doesn’t always happen.  Our school day starts at 7:25am so those 25 minutes can be pretty frantic.  Lunch goes into the fridge in the math office on my way to my room.  Then I walk into my classroom and wonder why I left it such a mess the afternoon before.  I then organize my paper stacks to give my desk some semblance of order.  I then need to run upstairs to grab copies, say hello to the office professionals up there, and hit the bathroom (very important).

We run a 5 period, trimester schedule, which makes each of our classes 68 minutes.  First hour is math recovery, which is about 20 kids that have failed a previous math course.  Their abilities are all over the place and their motivation to do mathematics is as well.  It’s a challenge to engage them sometimes, but they’ve gotten to know Andrew Stadel pretty well.  As Mr. Stadel talked about in his recent post, I too have acquired some puzzles over the course of the last 9 years and let this class work with them during the last 20 or so minutes of class.

Second hour is college algebra so I’ve got 7 minutes to run to the bathroom (I know but I drink a lot of soda).  This class is amazingly exhausting.  I’ve been just blown away by the mental power in that room.  When I give them a problem, those kids go AT IT.  I swear that the brain sweat is palpable.

Third hour is my prep period.  I have a million things going on during this 68 minutes since I am also the head of our department.  I never feel like I get enough done, but is there any teacher that feels like they are caught up ever?

We then have a 28-minute “study hall” time where kids can study, meet with clubs, go to the media center, make up assessments, etcetera.  Most students use the time to sit and chill out.  I don’t blame them; high school demands a lot.

At about 11:30, 4th hour officially begins.  This is an advanced probability and statistics course, and it has been fun to challenge these kids with real world scenarios.   These kids are naturally curious, and we often spend an entire class period discussing a problem and all of the statistics that come into play.  I end that class period wishing I taught it more than once a day.

It’s now 12:41.  Yep, the day starts at 7:25 and some of the kids don’t eat until 12:41.  Brutal.  I run to the bathroom, suck down my kale salad, and laugh with my co-workers.  One great thing about my department is lunch usually brings about hilariousness for one reason or another.   I really enjoy that comradery and I’m very lucky to be part of a department that enjoys one another.

1:17 is the beginning of 5th hour, so I will of course need to run to the bathroom one last time.  This class is college algebra again.  This is the last hour of the day, so the kids are a little more energetic but no less mathematically clever.

The bell rings at 2:25 for the day.  Unless I have a meeting after school, I leave between 2:50 and 3:30.  I want to get my T25 workout done before I pick up my daughter from her school.  (T25 is a great workout program from Beachbody.  Best workout I’ve ever done.)

Once my daughter is home, we usually build with Legos, play a game, or make a fort.  Bedtime routine starts at 6:00 so it’s not long before we are watching her allotted half-hour of TV and then reading books.  (I try to suggest Team Umizoomi everyday, but their mighty math powers are usually trumped by Minnie and her Bowtique.)

She’s asleep by about 7:30pm and we won the kid sleeping lottery, so unless it is an extreme circumstance, we don’t hear from her until we wake her in the morning.  Now it’s time to catch up on blogs, lesson plans, working through problems, or adding student feedback.  Since mathematics has also become a personal passion of mine, working on these things at night is enjoyable as well as productive.

I enjoy most of my days because I have a pretty positive outlook on being a teacher.  I have a strong belief that what I do makes a difference and that belief is what drives my passion for teaching.  When you show kids that you believe in them, there is a tremendous benefit to both you AND them.  Each day, I try to spread that to my students.  If they know that I believe in them first, they are more likely to believe in themselves and achieve more mathematically.

Full Circle Reflection

It’s almost the end of the trimester already which made today my last official “teaching” day with my Algebra class.  I’ve used a lot of the Math Forum’s Problems of the Week in this class.  Since this is a college algebra class, I use the POWS more as problems of the day rather than the week.  As a member, I have access to the library of problems, which I scour quite frequently to find just the right problem to fit the topic at hand.

Today’s adventure:  Rational Functions

I used a POW in which the first four terms of a patterned sequence of A’s and B’s are shown.  The students are asked to create an expression to represent the number of B’s in the nth term and then create an expression to represent the ratio of B’s to the total number of letters in the nth term.  What I like about this task in particular is that it isn’t a completely obvious fraction-ladened, asymptote-wielding, makes-a-student-want-to-cry rational function.  The students are able to work through most of the problem forgetting that this is in fact THAT type of function.  In fact, since they weren’t immediately scared off with a 1/x or the like, it seemed easier for them to make connections from their solutions to the graph and equation of the function.

What was particularly fantastic about this problem was that the growth of these students in the problem solving process was so evident.  It was clear as I circulated the room that over the course of a trimester, these students’ goals as mathematicians were evolving:  from “fast and correct” to “patient and curious.”

For example, when asked to find the term that results in 35% B’s, I had many students make a table with # of A’s, B’s, Total letters, and ratio of B’s to Total letters.  At the beginning of the trimester, these kids would accept their correct answer, but then reject their method of arriving at the answer because it was not as quick as those able to recall an equation or procedural method.  Now, after 13 weeks, these same kids are able to look at their table and appreciate the extra questions they can now address about this pattern scenario.  Additionally, some students were willing to attempt multiple methods in arriving at the answer.    It was a pretty profound moment for them as problem solvers and me as an algebra teacher.  I don’t know who was more proud, them or me.

Here are some samples of their work:

 

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Creating Mathematically Curious Students

Part of my goal for my students this year is to help them become comfortable being mathematically curious.  In an effort to help students develop a growth mindset and to facilitate learning opportunities that foster this, I try to pose questions that allow time for exploration.  This has been more difficult than I thought it would be, since I am somewhat attempting to undo 10-11 years of an unwritten didactic contract:  teachers instruct, students passively absorb and regurgitate information.   Repeat as many times as necessary.

Most of the trimester I have gotten questions like, “Is this what you are looking for?” or “Are we going to be tested on this?” I thought that as the trimester progressed, this would happen less often.  I was wrong.  I had to admit to myself that as long as I was in charge of giving these kids a grade and as long as their grades remained a driving force in their outlook on education, I didn’t see this changing much.

I wanted to make note, however, of the progress these kids have made toward being more mathematically curious.  I’ve exposed them to some interesting graphs, and some students have shared how they expanded those ideas to make even more intricate versions of those graphs.   (Ever wondered what y=xsin(1/x^13) looks like?)

I had another student explore the graph of y = 1/(x-2) + 1/(x-2) and wonder if there was an equation we could write that would isolate just the middle portion, between the two vertical asymptotes.  He thought that it looked cubic, so he played around with a number of cubic functions, but couldn’t get the graph to fit quite right.  I commended him for his efforts, because this was the type of student known for wanting his math straight to the point.  I asked him the range of a cubic function compared to the range of the graph he was trying to match, and he quickly saw that his initial thoughts on a cubic function were incorrect.  I challenged him to keep searching for an equation (or two) that would match the portion of the graph he wished to isolate.  This was an important moment for both me and this student.  I had gotten him to explore and wonder with something that had no external purpose.  He did this for the meer wonderment of whether it would work or not.

This was fantastic and worth reflecting over for me as an algebra teacher.  Much of high school algebra is taught in a dry, procedural manner.  Unfortunately, the kids expect it this way, and the high achieving kids even want it this way.  They’ve been successful with it so far, why change it?  I hope as I continue to pose mathematical questions to these kids that they continue to push their understanding forward by exploring.

Nrich – Factors and Multiples Puzzle

Nothing gets me more excited about teaching mathematics than a task that can engage my lower level students while simultaneously challenge my high achieving students. The Factors and Multiples Puzzle from Nrich did just that. (Thanks to @drrajshah for posting this on twitter.)

I’m glad I used this in multiple classes because if nothing else, it gave students the opportunity to learn about triangular numbers! What a testament to the fact that we don’t allow students to explore with numbers nearly enough: I’ll bet only one student out of 60 had any idea what triangular numbers were.  A fantastic, interesting set of numbers, arithmetically and visually, was unbeknownst to 99% of my students.

My math recovery students were intrigued by the puzzle portion of it. In fact, I have one student in particular who is not particularly motivated by much . He’s a ‘too cool for school’ kind of kid, and he’ll tell you as much. When I bust out a puzzle, he’s all in. And when I say ‘all in,’ I mean 100%, until he solves it. It’s pretty awesome stuff to have been able to catch his attention and see how cleverly he thinks through things. Amazing.

I also gave this task to a group of advanced students. An interesting strategy these students developed was to grab a whiteboard to work out some patterns in groups of numbers.  I loved walking around and hearing their strategies.  As some groups finished, they started walking around and giving tips (not answers) to other groups.  It was wonderful.

One of my particularly eager students taped his together uniquely.  I appreciated his humor.  🙂

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Why I Blog, for Kate

Kate Nowak was the one, whether she knows it or not, that gave me the convincing boost I needed to start blogging.  So I figured I owed it to her to respond to her request of “why I blog.”

1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?

I was taking a PD class called Thinking Mathematics through my districts Teacher Academy.  As part of the course, we were to read a chapter from Accessible Mathematics.  Of course, I bought the whole book instantly because it was exactly what I needed to get me excited about changing some of my teaching practices to become a better teacher. The book was so easy to read, easy to follow and made so much sense.

Anyway, as part of this class, we were given time to develop a unit plan and formulate lessons.  I was determined to scour the internet for some good resources.  I found myself flooded with them.  I can’t recall the very first blog I came across, but I found that I needed to start reading blogs regularly because there were some great math teacher bloggers with some great ideas and who were open and freely willing to share their resources.  I was immediately hooked.  Good thing I already had an iphone.
2. What keeps you coming back? What’s the biggest thing you get out of reading and/or commenting?

I found that the more you give, the more you get.  I started blogging and commenting on blogs at about the same time.  The more I commented, the more I wanted to write more blog posts, the more I wanted to comment, the more I wanted to blog, the more I …you get the idea.

I am find that I always love to hear other people’s ideas face to face.  I knew that reading about other people’s ideas could be even more fun!

Just as students learn more about mathematics by talking to one another about mathematics,  we as teachers should take that same advice.  The more we collaborate across the web, the more multi-faceted our lessons can be.
3. If you write, why do you write? What’s the biggest thing you get out of it?

Enter Kate Nowak.  Once at a Global Math Department meeting, she mentioned something about why SHE started blogging.  She said she started blogging for herself, to get her teaching ideas out of her brain and to reflect on her lessons.  I took this to heart because I realized that if I was going to blog, my goal should be for self-reflection.

The first blog post I wrote was for Dan Meyer’s Makeover Monday.  It was the last week where he kills it with the Desmos Penny Circle.  I was very intrigued that there were other teachers across the country that cared about my input on a particular task.  It shouldn’t have been a surprise to me.  I care what other teachers have to say, why shouldn’t other teachers care what I think?
4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to’s? Stories?

Your story.  How blogging transformed your teaching and your view of how teachers connect.  And how easy it is to get started.  I’d love to hear it.  Good luck, Kate.

Questimate saves the day!

My computer was being upgraded to Windows 8, so I didn’t have a computer for 1st hour.  My first hour is my math recovery class (i.e. students who had failed a previous high school math class.)  We usually start out doing Visual Patterns or Estimation 180, so I hoped that I was able to keep the first part of the class productive.

Enter Questimate.  If you have never seen this amazing, engaging, fascinating ipad app before, you are in for a treat.  Made by Motion Math, the players come up with their own estimation questions from a list of choices.  For example, how many blue whale tongues are as heavy as a Marlin?  Or:Questimate! Make your question!

My favorite is when they can size the objects themselves:

Questimate! Visual pinchingThe kids in this math class are not thrilled with school in general, as you can imagine.  They were all totally engaged with Questimate.  I gave each kid a chance to create a question and estimate an answer.  Happy kids, Happy teacher. It was a great class period.